Problem 49 - Calculus with Concepts in Calculus

Question

A single respiratory cycle includes one inhalation and one exhalation. During one respiratory cycle of a certain person at rest, the rate of flow \(R\) (in liters per second) of air into a person's lungs at time \(t\) (in seconds) is given by $$ R=0.5 \sin \frac{2 \pi}{5} t $$ a. How long does it take to complete one respiratory cycle? b. How many respiratory cycles are completed in one minute? c. Graph one complete cycle, starting at time \(t=0\). d. Interpret the meaning of positive and negative values of \(R\) e. To the nearest hundredth, find \(R\) when \(t=3\) seconds.

Step-by-Step Solution

Answer

a) 5 seconds; b) 12 cycles; c) Graph shows one period of a sine wave; d) Positive \( R \) is inhalation, negative \( R \) is exhalation; e) \( R \approx 0.47 \).

Step 1: Understanding the Period of the Sine Function

The function given is \( R = 0.5 \sin\left(\frac{2\pi}{5}t\right) \). For a sine function of the form \( R = A \sin(Bt) \), the period \( T \) is calculated as \( T = \frac{2\pi}{B} \). In our function, \( B = \frac{2\pi}{5} \), so the period \( T = \frac{2\pi}{\frac{2\pi}{5}} = 5 \) seconds. This means one complete respiratory cycle takes 5 seconds.

Step 2: Calculating Respiratory Cycles Per Minute

Since one respiratory cycle takes 5 seconds, the number of cycles in one minute (60 seconds) can be calculated using the conversion: \[\text{Number of cycles per minute} = \frac{60}{5} = 12.\]

Step 3: Graphing One Complete Cycle

To graph one complete cycle of \( R = 0.5 \sin\left(\frac{2\pi}{5} t\right) \) from \( t=0 \) to \( t=5 \), we observe:- At \( t=0 \), \( R = 0 \).- At \( t=1.25 \), \( R \approx 0.5 \) (the peak value).- At \( t=2.5 \), \( R = 0 \).- At \( t=3.75 \), \( R \approx -0.5 \) (the trough value).- At \( t=5 \), \( R = 0 \).This creates a sinusoidal wave between 0 and 0.5, peaking at 1.25 seconds and troughing at 3.75 seconds.

Step 4: Interpretation of Positive and Negative Values of R

In the context of the respiratory cycle, positive values of \( R \) indicate inhalation, where air is flowing into the lungs. Negative values of \( R \) indicate exhalation, where air is flowing out of the lungs.

Step 5: Calculating R at t=3

To find \( R \) at \( t=3 \) seconds, substitute this value into the function: \[R = 0.5 \sin\left(\frac{2\pi}{5} \times 3\right)\]\[= 0.5 \sin\left(\frac{6\pi}{5}\right)\]The angle \( \frac{6\pi}{5} \) is equivalent to \( \frac{\pi}{5} \) radians beyond a full rotation \((2\pi)\), so we use the sine of this angle:\[R = 0.5 \sin\left(\frac{\pi}{5}\right) \approx 0.47.\]

Key Concepts

Sine Function

A sine function is a mathematical function that creates a smooth, wave-like pattern, commonly associated with periodic phenomena like sound, light waves, or even respiratory cycles. The general form of a sine function is expressed as \( R = A \sin(Bt) \), where:

• \( A \) represents the amplitude, which is the peak value the function reaches. In the context of a respiratory cycle, it reflects the maximum rate of air flow into the lungs.
• \( B \) affects the period of the function, determining how quickly it repeats its cycle.

By manipulating these parameters, you can model various periodic behaviors. In our case, the function \( R = 0.5 \sin\left( \frac{2\pi}{5} t \right) \) models the flow rate of air during respiratory cycles, capturing both inhalation and exhalation phases. Understanding how the sine function behaves helps us accurately describe these cycles.

Graphing

Graphing a sine function allows us to visualize the oscillatory nature of the cycle. When plotting \( R = 0.5 \sin\left( \frac{2\pi}{5} t \right) \) on a graph, the curve oscillates between the values of -0.5 and 0.5.

• At \( t=0 \) and \( t=5 \), the graph starts and ends at zero, indicating no air flow, marking the transition between inhalation and exhalation.
• The peak inhalation occurs at \( t=1.25 \) seconds, where the graph reaches its maximum value, \( R = 0.5 \).
• Exhalation is strongest at \( t=3.75 \) seconds, as the graph dips to its minimum value of \( R = -0.5 \).

This sinusoidal shape is typical for functions modeling periodic events, making it an effective visual representation for understanding when and how strongly air flows into and out of the lungs.

Periodicity

Periodicity is a fundamental concept referring to the repetition of a cycle at regular intervals. In the sine function \( R = 0.5 \sin\left( \frac{2\pi}{5} t \right) \), periodicity is determined by the factor \( B \) in \( \frac{2\pi}{B} \), which calculates the period. In our respiratory cycle example, the period \( T \) is:\[T = \frac{2\pi}{\frac{2\pi}{5}} = 5 \text{ seconds}\]This means each complete cycle of inhalation and exhalation takes 5 seconds.

• This consistent timing allows us to see how often these cycles repeat, which is crucial, for instance, in measuring cycles per minute and confirming regular breathing patterns.
• Understanding periodicity ensures that models can predictively mirror real-world processes, like the rhythmic nature of breathing.

Trigonometric Functions

Trigonometric functions describe relationships between angles and side lengths in right triangles, and they extend to periodic phenomena like cycles or waves.

• The sine function is one among several trigonometric functions, specifically modeling cyclical patterns through its smooth oscillation.
• In the formula \( R = 0.5 \sin\left( \frac{2\pi}{5} t \right) \), the sine function captures how air flow changes rhythmically, with amplitude indicating flow strength and \( B \) defining the timing of the cycle.

These mathematical tools are invaluable in disciplines like physics, engineering, and many branches of science, offering a precise method to model and analyze repetitive and wave-like behavior in systems ranging from electronics to biology.